We present a detailed analysis of the Landau-Zener problem for an interacting Bose-Einstein condensate in a time-varying double-well trap, especially focusing on the relation between the full many-particle problem and the mean-field approximation. Due to the nonlinear self-interaction a dynamical instability occurs, which leads to a breakdown of adiabaticity and thus fundamentally alters the dynamics. It is shown that essentially all the features of the Landau-Zener problem including the depletion of the condensate mode can be already understood within a semiclassical phase-space picture. In particular, this treatment resolves the formerly imputed incommutability of the adiabatic and semiclassical limits. The possibility of exploiting Landau-Zener sweeps to generate squeezed states for spectroscopic tasks is analyzed in detail. Moreover, we study the influence of phase noise and propose a Landau-Zener sweep as a sensitive yet readily implementable probe for decoherence, since the noise has significant effect on the transition rate for slow parameter variations.